Rigorous derivation of the Landau equation in the weak coupling limit

TL;DR

This paper rigorously derives the linear Landau equation as the weak coupling limit of microscopic plasma models, unifying previous results across a range of parameters and analyzing the diffusion coefficient.

Contribution

It extends and unifies prior work by showing negligible clusters of obstacles in the limit and proves the diffusion coefficient is independent of the parameter.

Findings

Microscopic models converge to the Landau equation in the weak coupling limit.

Clusters of overlapping obstacles are negligible in the limit.

The diffusion coefficient is independent of the parameter.

Abstract

We examine a family of microscopic models of plasmas, with a parameter $\alpha$ comparing the typical distance between collisions to the strength of the grazing collisions. These microscopic models converge in distribution, in the weak coupling limit, to a velocity diffusion described by the linear Landau equation (also known as the Fokker-Planck equation). The present work extends and unifies previous results that handled the extremes of the parameter $\alpha$, for the whole range (0, 1/2], by showing that clusters of overlapping obstacles are negligible in the limit. Additionally, we study the diffusion coefficient of the Landau equation and show it to be independent of the parameter.